Congruences for powers of the partition function

Madeline Locus, Ian Wagner

Published 2016-04-26Version 1

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such congruences, those of the form $p_{-t}(\ell n + a) \equiv 0 \pmod {\ell}$, were previously studied by Kiming and Olsson. If $\ell \geq 5$ is prime and $-t \not \in \{\ell - 1, \ell -3\}$, then such congruences satisfy $24a \equiv -t \pmod {\ell}$. Inspired by Lin's example, we obtain natural infinite families of such congruences. If $\ell\equiv2\pmod{3}$ (resp. $\ell\equiv3\pmod{4}$ and $\ell\equiv11\pmod{12}$) is prime and $r\in\{4,8,14\}$ (resp. $r\in\{6,10\}$ and $r=26$), then for $t=\ell s-r$, where $s\geq0$, we have that \begin{equation*} p_{-t}\left(\ell n+\frac{r(\ell^2-1)}{24}-\ell\Big\lfloor\frac{r(\ell^2-1)}{24\ell}\Big\rfloor\right)\equiv0\pmod{\ell}. \end{equation*} Moreover, we exhibit infinite families where such congruences cannot hold.